248 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF spar or other crystal, cut for shewing rings, followed by a Nicol's i prism. The plane-polarized pencils respectively stopped by and 1 transmitted through the Nicol's prism consisted, on entering the crystal, of pencils elliptically polarized in opposite ways; and i the nature of this elliptic polarization changes in every possible i ' manner from one point to another of the field of view. If these i two streams of light be equivalent according to the definition given in the preceding article, they will present exactly the same appearance on being viewed through a crystal followed by a Nicol's prism or other analyzer. 11. THEOREM. Let a polarized stream be resolved into two 1 j oppositely polarized streams; let the phase of vibration of one I M ; of the streams be altered by a given quantity relatively to that I1 j i of the other, and let the streams be then compounded. If the I : If polarization of the original stream be now changed to its oppo-I i) site, the polarization of the final stream will also be changed to 1 I! , its opposite. *f 11 i The straightforward mode of demonstrating this theorem, by !f ! making use of the general expressions, would lead to laborious f \ analytical processes, which are wholly unnecessary. For the l| *' formulae which determine the components of a given stream are | I expressed by simple equations, so that the results are unique, and '/, • accordingly whenever we can foresee what the result will be, it is | , sufficient to shew that the formulas themselves, or the geometrical j * (! conditions of which the formulse are merely the expressions, are . ( , satisfied. 1 i' For shortness' sake call the original stream X, and its com- , \'! ponents 0, E. Let p be the given quantity, positive or nega- tive, by which the phase of vibration of 0 is retarded relatively to that of E. Let o, e denote the streams 0, E after the changes of phase, and Y the stream resulting from their reunion. Con-, ceive now all the vibrations with which we are concerned to be turned in azimuth through 90°. This will not affect the geometrical relations connecting components and resultants. Let X', 0', E', o'', e, Y be the streams which X, 0, E, o, e, Y thus become. The streams 0'', E' are evidently polarized in the same manner respectively as E, 0, except that right-handed is changed into left-handed, and vice versa; and in passing from 0' to c/ the ; phase is retarded by p. Now conceive the direction of motion of